Click "Save private key". This file will be like your password, so place it somewhere safe (not in a p2p shared folder for example).
You can also save your public key somewhere if you want it for future reference.

Close PuTTYgen.

Run Pageant

Rightclick the little computer with the hat in the lower right corner of the screen.

Click "Add key".

Browse to your saved private key file (the .ppk).

Enter your passphrase when prompted. If you forgot your passphrase you'll have to go back to step 2 and generate a new key.

Now try to checkout with TortoiseSVN using the development url from your project page.
If it isn't working make sure you did every step right. If it still isn't working, take a note of the error message you're getting and comment here or ask in the CurseForge IRC channel on freenode.org.

A better way to start Pageant

Pageant will not remember your key the next time you run it so you will have to add your key every time you start it.
You can do this by making a shortcut with target like this:

C:\path\to\pageant.exe "C:\path\to\private_key.ppk"

It will still ask you for the passphrase but at least you don't have to browse for the key file.
You can put the shortcut in the autostart folder if you want and it will automaticly run when you startup.